Riemannian manifold learning has regained attention due to appealing geometric properties that allow methods to represent non-Euclidean data arising in several domains [1]}.
We propose matrix models of Hermitian symmetric spaces, which are complex manifolds modelled on \(, to learn representations for undirected graphs. Our approach leverages the rich structure of these manifolds that combine aspects of Euclidean and hyperbolic geometry. We develop and implement tractable and mathematically sound algorithms to learn embeddings through gradient-descent methods.\footnote {Code available at: \url {https://github.com/fedelopez77/sympa}}We suggest a new class of synthetic datasets for benchmarking graph embeddings. They are constructed as rooted products of grids and trees, exhibiting grid- and tree-like structures at different scales.We showcase the effectiveness of the proposed approach on conventional as well as new datasets for the graph reconstruction task.Our method ties or outperforms constant-curvature and Cartesian product baselines without requiring any previous assumption on the geometric features. This shows the flexibility and enhanced representation capacity of Hermitian symmetric spaces.\) As future directions, we consider extending the distance computation algorithms to Finsler metrics, developing the missing components for other optimization techniques, such as Riemannian Adam [2]}, further experimentation and analysis of the current models, and the application of the proposed techniques to a structural analysis of graphs as well as to downstream tasks.
